--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/stackTest.agda	Wed Jan 10 15:44:13 2018 +0900
@@ -0,0 +1,128 @@
+open import Level renaming (suc to succ ; zero to Zero )
+module stackTest where
+
+open import stack
+
+open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.Core
+open import Data.Nat
+
+
+open SingleLinkedStack
+open Stack
+
+----
+--
+-- proof of properties ( concrete cases )
+--
+
+test01 : {n : Level } {a : Set n} -> SingleLinkedStack a -> Maybe a -> Bool {n}
+test01 stack _ with (top stack)
+...                  | (Just _) = True
+...                  | Nothing  = False
+
+
+test02 : {n : Level } {a : Set n} -> SingleLinkedStack a -> Bool
+test02 stack = popSingleLinkedStack stack test01
+
+test03 : {n : Level } {a : Set n} -> a ->  Bool
+test03 v = pushSingleLinkedStack emptySingleLinkedStack v test02
+
+-- after a push and a pop, the stack is empty
+lemma : {n : Level} {A : Set n} {a : A} -> test03 a ≡ False
+lemma = refl
+
+testStack01 : {n m : Level } {a : Set n} -> a -> Bool {m}
+testStack01 v = pushStack createSingleLinkedStack v (
+   \s -> popStack s (\s1 d1 -> True))
+
+-- after push 1 and 2, pop2 get 1 and 2
+
+testStack02 : {m : Level } ->  ( Stack  ℕ (SingleLinkedStack ℕ) -> Bool {m} ) -> Bool {m}
+testStack02 cs = pushStack createSingleLinkedStack 1 (
+   \s -> pushStack s 2 cs)
+
+
+testStack031 : (d1 d2 : ℕ ) -> Bool {Zero}
+testStack031 2 1 = True
+testStack031 _ _ = False
+
+testStack032 : (d1 d2 : Maybe ℕ) -> Bool {Zero}
+testStack032  (Just d1) (Just d2) = testStack031 d1 d2
+testStack032  _ _ = False
+
+testStack03 : {m : Level } -> Stack  ℕ (SingleLinkedStack ℕ) -> ((Maybe ℕ) -> (Maybe ℕ) -> Bool {m} ) -> Bool {m}
+testStack03 s cs = pop2Stack s (
+   \s d1 d2 -> cs d1 d2 )
+
+testStack04 : Bool
+testStack04 = testStack02 (\s -> testStack03 s testStack032)
+
+testStack05 : testStack04 ≡ True
+testStack05 = refl
+
+------
+--
+-- proof of properties with indefinite state of stack
+--
+-- this should be proved by properties of the stack inteface, not only by the implementation,
+--    and the implementation have to provides the properties.
+--
+--    we cannot write "s ≡ s3", since level of the Set does not fit , but use stack s ≡ stack s3 is ok.
+--    anyway some implementations may result s != s3
+--  
+
+stackInSomeState : {l m : Level } {D : Set l} {t : Set m } (s : SingleLinkedStack D ) -> Stack {l} {m} D {t}  ( SingleLinkedStack  D )
+stackInSomeState s =  record { stack = s ; stackMethods = singleLinkedStackSpec }
+
+push->push->pop2 : {l : Level } {D : Set l} (x y : D ) (s : SingleLinkedStack D ) ->
+    pushStack ( stackInSomeState s )  x ( \s1 -> pushStack s1 y ( \s2 -> pop2Stack s2 ( \s3 y1 x1 -> (Just x ≡ x1 ) ∧ (Just y ≡ y1 ) ) ))
+push->push->pop2 {l} {D} x y s = record { pi1 = refl ; pi2 = refl }
+
+
+id : {n : Level} {A : Set n} -> A -> A
+id a = a
+
+-- push a, n times
+
+n-push : {n : Level} {A : Set n} {a : A} -> ℕ -> SingleLinkedStack A -> SingleLinkedStack A
+n-push zero s            = s
+n-push {l} {A} {a} (suc n) s = pushSingleLinkedStack (n-push {l} {A} {a} n s) a (\s -> s ) 
+
+n-pop :  {n : Level}{A : Set n} {a : A} -> ℕ -> SingleLinkedStack A -> SingleLinkedStack A
+n-pop zero    s         = s
+n-pop  {_} {A} {a} (suc n) s = popSingleLinkedStack (n-pop {_} {A} {a} n s) (\s _ -> s )
+
+open ≡-Reasoning
+
+push-pop-equiv : {n : Level} {A : Set n} {a : A} (s : SingleLinkedStack A) -> (popSingleLinkedStack (pushSingleLinkedStack s a (\s -> s)) (\s _ -> s) ) ≡ s
+push-pop-equiv s = refl
+
+push-and-n-pop : {n : Level} {A : Set n} {a : A} (n : ℕ) (s : SingleLinkedStack A) -> n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack s a id) ≡ n-pop {_} {A} {a} n s
+push-and-n-pop zero s            = refl
+push-and-n-pop {_} {A} {a} (suc n) s = begin
+   n-pop {_} {A} {a} (suc (suc n)) (pushSingleLinkedStack s a id)
+  ≡⟨ refl ⟩
+   popSingleLinkedStack (n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack s a id)) (\s _ -> s)
+  ≡⟨ cong (\s -> popSingleLinkedStack s (\s _ -> s )) (push-and-n-pop n s) ⟩ 
+   popSingleLinkedStack (n-pop {_} {A} {a} n s) (\s _ -> s)
+  ≡⟨ refl ⟩
+    n-pop {_} {A} {a} (suc n) s
+  ∎
+  
+
+n-push-pop-equiv : {n : Level} {A : Set n} {a : A} (n : ℕ) (s : SingleLinkedStack A) -> (n-pop {_} {A} {a} n (n-push {_} {A} {a} n s)) ≡ s
+n-push-pop-equiv zero s            = refl
+n-push-pop-equiv {_} {A} {a} (suc n) s = begin
+    n-pop {_} {A} {a} (suc n) (n-push (suc n) s)
+  ≡⟨ refl ⟩
+    n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack (n-push n s) a (\s -> s))
+  ≡⟨ push-and-n-pop n (n-push n s)  ⟩
+    n-pop {_} {A} {a} n (n-push n s)
+  ≡⟨ n-push-pop-equiv n s ⟩
+    s
+  ∎
+
+
+n-push-pop-equiv-empty : {n : Level} {A : Set n} {a : A} -> (n : ℕ) -> n-pop {_} {A} {a} n (n-push {_} {A} {a} n emptySingleLinkedStack)  ≡ emptySingleLinkedStack
+n-push-pop-equiv-empty n = n-push-pop-equiv n emptySingleLinkedStack